In survey applications using GNSS equipment, the measurement taken by the GNSS receiver does not directly correspond to the target point to be surveyed; rather, it is a position measurement of the so-called “phase-centre” of the GNSS antenna. Current practice is to mount the antenna on a pole and ensure the pole is perfectly vertical over the point of interest and compensate the position for the length of the pole. Levelling the pole, however, takes time and it would be desirable to take measurements of the target position without the need to level the pole.
In this document, a method is detailed for the accurate compensation of the tilt of the pole using a fusion of relatively low-cost inertial sensors (accelerometers and gyroscopes) and GNSS measurements. It has advantages over other disclosed methods as the inertial sensors are of substantially lower cost and sensor errors are explicitly estimated at run-time.
Global Navigation Satellite Systems (GNSS), such as the NAVSTAR Global Positioning System (GPS) are now routinely used in surveying applications with the use of Real-Time Kinematic (RTK) algorithms, which correct for a receiver's position using information from one or more nearby base station(s).
The position measured by the GNSS receiver is at the so-called “phase centre” of the GNSS antenna, which for a high-quality survey antenna is a well-quantified location generally near the mechanical centre of the antenna. However, the location of interest to the surveyor is not at the antenna, but at a point on the ground; the antenna is usually mounted on a pole to optimise the reception of GNSS signals.
Unless the pole is perfectly upright with respect to the ground, the horizontal position of the antenna will be offset compared to the location on the ground of interest. A levelling device, whether a traditional spirit level or a more sophisticated arrangement of sensors, can be used to determine if a pole is upright to some degree of tolerance sufficient to take a measurement. Alternatively, if one can precisely measure the angular orientation (the “attitude”) of the pole, this error can be compensated for, assuming one knows the length of the pole.
Measuring the angle of the pole with respect to the ground is not a trivial exercise. Whilst measuring the angle of the pole from the vertical (the pitch and roll) can be accurately achieved through a number of methods (notably, measurement of local gravity using accelerometers or inclinometers), measuring the orientation of the pole with respect to True North (the yaw or azimuth) is considerably more difficult.
The most obvious way to achieve a measurement of azimuth is with the use of an electronic compass, which is able to measure orientation with respect to Magnetic North. However, aside from the offset between Magnetic North and True North, the compass reading can also be affected by magnetic field disturbances such as ferrous metals and electric currents, both of which are common around some building sites. To avoid these drawbacks, another method of determining the yaw is desirable.
When installed in a conventional land vehicle such as a car, the azimuth angle is able to be inferred from the GNSS velocity, since the vehicle is normally aligned with the direction of travel. However, since a pole-mounted antenna may move in an arbitrary direction, GNSS velocity is not a reliable means of azimuth determination.
When stationary, high-grade inertial sensors are capable of measuring the Earth's rotation rate, which may then be used for finding north. When using high-quality gyroscopes (“gyros”), a procedure known as “gyro compassing” can be used whilst stationary to determine north by comparing the measured rotation rate in each axis.
Once an initial position and attitude is known, a high-quality INS can navigate without reference to GNSS or other external measurements, making it useful for survey applications when GPS is unavailable. This is described, for instance in US 2009 024 325 A1 which describes an INS used in a surveying application with a GNSS solution being unavailable. However, as an inertial navigation solution mathematically rotates and integrates the raw inertial sensor measures, small errors will accumulate and the position solution (and azimuth solution, which must be maintained as a by-product) will drift without bound.
The drift problem of an INS has been studied extensively for some time. The drift in the position and attitude solution (and thereby the means to compensate for pole tilt) can be contained by intelligently fusing GNSS or other external measurements with an inertial navigation system—the so-called “Aided Inertial Navigation System” (AINS)—which has long existed in the aerospace industry and is now used in surveying applications.
Whilst high-quality “navigation grade” and “tactical grade” inertial sensors—navigation grade refers to sensors that can be useful for standalone navigation for many hours, e.g. including Ring Laser Gyroscopes (RLG), whereas tactical grade generally refers to navigation requirements for short flights, e.g. including Fibre Optic Gyroscopes (FOG)—are undoubtedly useful in determining position and attitude (whether GNSS aided or otherwise), they are also very expensive, heavy, bulky and suffer from high power consumption. In contrast, the last decade or so has seen the rise of inertial sensors based on MEMS (Micro-Electro-Mechanical Sensors) technology, which have substantially lower performance in comparison with traditional high-grade devices, but have the advantage of being small, low-power lightweight, and more than an order of magnitude less expensive than high-grade inertial sensors. MEMS inertial sensors are now routinely integrated into low-accuracy consumer applications such as games console controllers and mobile phones. MEMS sensors are normally of “consumer grade” or “industrial grade”, though some high-end silicon MEMS may also be considered as tactical grade. Consumer grade refers to cheap sensors having coarse motion sensing for applications such as shock detection, free-fall detection, mobile phones or computer games controllers. Industrial grade refers to sensors that are useful for applications where some degree of motion sensing precision is required, such as manufacturing robots, machine control, automotive electronic stability, hill-start assistance and entry-level Attitude and Heading Reference Systems (AHRS).
MEMS sensors, like many integrated circuit technologies, have substantially improved in performance over time. Although they presently remain unsuitable for standalone inertial navigation, they may be fused with GNSS measurements in a similar fashion to a high-grade AINS solution to maintain an attitude solution of sufficient accuracy to compensate for the tilt of a survey pole. Furthermore, the combination of GNSS and INS is greater than the sum of its parts-intelligent integration of the two allow for the most substantial errors that exist in MEMS inertial sensors to be estimated and removed.
The drawback of using MEMS devices is that it relies on a good-quality GNSS solution to be available, which is generally the case for many surveying and related activities such as stake-out. Usually, more than a few seconds without a GNSS solution will cause the attitude to drift out of tolerance, depending on the grade of sensor. Generally, a high-quality position is required survey applications and therefore attitude drift during GNSS outages is usually not problematic. Once a GNSS reacquired, a smaller drift will speed up re-convergence of the attitude solution. It is up to the designer to make the trade-off between stability and cost.
Attitude may be interpreted as a combination of three different rotations-roll, which (when related to an aircraft) is “wings up, wings down”; pitch, which is “nose up, nose down” and yaw, which corresponds to the heading that the platform is pointing. Classically, pitch and roll are observed through measurements of the local gravity vector (which induce an acceleration measurement on the accelerometers) and yaw is observed through the use of a magnetic compass.
Whilst in prior art—for example in US 2003 046 003 A1, U.S. Pat. No. 5,512,905 A, EP 2 040 029 A1, EP 1 726 915 A1 and JP 2005 043 088 A—an accelerometer (“tilt sensor”) and compass has been used previously to compensate for the tilt of a pole, it is limited by the accuracy of the sensors and subject to local magnetic field disturbances. For example, a typical MEMS accelerometer may be four degrees or more in error without careful factory calibration, which is unacceptable for survey applications. Furthermore, without a high-fidelity and computationally expensive magnetic model, even the local declination angle (the angle between true north and magnetic north) may be in error by up to three degrees, even without any local disturbing fields caused by (for example) cars and power lines.
When inertial sensor measurements are intelligently combined with GPS, both the yaw angle (even without the aid of a magnetic compass) and sensor errors causing the errors in pitch and roll may be estimated when subjected to particular motion conditions. In particular yaw angle, generally considered to be the hardest to estimate, has been shown in the academic literature to converge towards its true value when the inertial sensors are subjected to changes in acceleration in the horizontal plane, which is often the case during a surveying operations. The situation is further complicated for MEMS devices, which have significant errors compared to high-grade devices. In high-grade sensors, bias errors on the vertical gyro are naturally observable through the rotation of the Earth. In MEMS devices, the signal caused the Earth's rotation is buried in noise and bias. Therefore, MEMS will require more aggressive manoeuvring to make yaw observable, but the basic fact that it is observable under motion is unchanged.
With this observation in mind, it can be noted that a magnetic compass is not strictly necessary for the estimation of yaw (and therefore can be removed if cost is a concern), but may still be used for a coarse initialisation of yaw or to provide supplementary measurements if the yaw has drifted due to the lack of motion for some time.
The notation used in this document is as follows:
An identity matrix of size k is denoted by Ik. The rotation matrix from the a-frame to the b-frame is denoted by Rab.
αabc denotes a vector quantity α of the b-frame with respect to the a-frame, expressed in terms of the c-frame.
[A]x is a skew-symmetric matrix constructed from vector A such that when multiplied by vector B the result is equivalent to the cross-product of A and B, viz [A]xB=A×B.
Time derivatives of a quantity are expressed using the dot notation (e.g. {dot over (ω)}abc), estimated quantities (as distinct from their true values) are denoted with a hat (e.g. {circumflex over (ω)}abc).
In this document, the b-frame is the body fixed frame, the n-frame is an earth-fixed local tangent frame (north, east, down), the e-frame is the Earth-Centred Earth-Fixed (ECEF) frame and the i-frame is the Earth-Centred Inertial (ECI) frame.
The GPS/INS estimates a position at the reference point of INS, rebe. The position on the ground at the end of the pole, repe, is desired. Since the offset of the INS reference point from the tip of the pole, rbpb, is known (i.e. the pole length and the location of the INS reference point on the end of the pole), one may calculate the point at the tip of the pole using:repe=Rberbpb+rebe 
Key to the accurate compensation of tilt is the attitude estimate. Clearly, any errors in attitude will couple onto errors at the ground point.
The observation of attitude and sensor errors relies on comparing the difference between the GPS position and the INS position. GPS velocity may also be used. Under motion, position errors coupled in from attitude errors can be indirectly separated. Note that an INS must maintain an accurate representation of attitude as a by-product of calculating a position solution.
Integrated GPS/INS navigation systems have long been used in the aerospace domain, owing the complementary characteristics of the two navigation sources. As such, multiple textbooks have been published on the subject which recommend a so-called loosely-coupled error-state system, for instance Robert M. Rogers: Applied Mathematics in Integrated Navigation Systems. AIAA Education, 3rd edition, 2007.
There are several components in the integration architecture:    1. An Inertial Measurement Unit (IMU), comprising three accelerometers, three gyroscopes in a nominally orthogonal configuration, associated support circuitry for signal acquisition, pre-processing, time synchronisation and deterministic error removal;    2. The Inertial Navigation System (INS) mechanisation, which mathematically rotates and integrates the accelerations and rotation rate measurements from the IMU to estimate the position and attitude;    3. A GPS receiver, measuring position and velocity of the antenna;    4. A Kalman Filter, which estimates the errors of position, errors in velocity, errors in attitude, gyro biases and accelerometer biases, based on the error dynamics of the system and observed difference between the GPS and INS positions; and    5. Optionally, a two- or three-axis magnetometer for measuring the Earth's magnetic field.
The key feature of the classic loosely-coupled error-state estimator is the linearization of the dynamics, which allows the use of a linear Kalman Filter. The major drawback of using such an architecture is the assumption that the errors are small and hence the error introduced by linearization is negligible. Whilst this might be true using high-quality sensors and a specific initialisation procedure, surveying applications (and especially construction surveying) is significantly more cost sensitive than the aerospace domain and hence estimation methods designed for high-quality sensors are not necessarily applicable to lower-cost MEMS inertial sensors described in the introduction. In particular, the small error assumption results in poor performance when using low-cost sensors. It is inherently clear that a better estimator is required for use with low-cost sensors.
In particular, the standard methods of GPS/INS integration, as for instance disclosed in US 2009 024 325 A1, have insufficient performance to reliably determine the attitude of the pole when using industrial-grade sensors. Therefore, these methods are fundamentally not viable for industrial grade sensors.